Lecture 3 - Walks and Trails

Report
MCA 520: Graph Theory
Instructor
Neelima Gupta
[email protected]
Table of Contents
Walks, Trails and Paths
Walks
• May have repeated Edges and Vertices.
• In case of multi-graph, we include the edges
also. In a simple graph, we can omit the edges
and simply mention the sequence of vertices.
• Closed Walk
Trails
• No repeated edges but vertices may repeat.
• Closed Trail
Path
• Neither vertices nor edges repeat.
• Definition:
– we say that a u-v walk W contains a u-v path P if
all the edges and vertices of P occur in W and in
that order but not necessarily consecutive.
– Similarly a closed walk W contains a cycle C if …..
• Lemma: Every u-v walk contains a u-v path
Odd/Even Walk
• Odd/Even walk : number of edges is odd/even
• Lemma: Every closed odd walk contains an
odd cycle.
• Remark: A closed even walk need not even
contain a cycle, it may simply repeat edges.
But, if an edge e appears exactly once in a
closed walk, then the walk contains a cycle
through e.
Even Graph
• A vertex is stb even(/odd) if its degree is
even(/odd).
• A graph is stb an even graph if all its vertices
are even.
Maximal Path
• A path in a graph is stb maximal if it is not
contained in a longer path.
– If a graph is finite, maximal path always exists.
• If every vertex in a finite graph G has degree at
least 2 then it contains a cycle.
– This is not true if the graph is not finite.
Connection Relation
• (u,v): u is stb connected to v ……
• Symmetric, Reflexive, Transitive
• Equivalence Relation
• Equivalence Class: Connected Component
• Lemma: A graph with n vertices and k edges
has at least n – k components.
• Proof: A graph with no edges has n
components. Adding an edges reduces the
number of components by at most 1. Thus
after adding k edges, number of components
is at least n – k.
Deleting an edge/vertex
• G – e: Deleting an edge does not delete its incident vertices.
• G – v: Deleting a vertex delete its incident edges.
• Thus deleting an edge may increase the number of components by
at most 1.
• Deleting a vertex v may increase the number of components by
(more) at most deg(v) – 1.
• Induced Graph G[T] = Graph that remains after deleting some
vertices such that the set of remaining vertices is T. i.e.
G[T] = (T, E(T)), where E(T) = {(u, v):u,v are in T and (u,v) is an edge in G}
•
Every subgraph of a graph need not be an induced subgraph.
Cut-edge and Cut-Vertex
• An edge e is stb a cut edge if …
• A vertex v is stb a cut vertex if …
Characterize cut-edges in terms of cycles.
• Theorem: An edge is a cut edge iff it does not
belong to any cycle.
Bi-partite Graphs
• Konig Theorem : Characterizing Bipartite
Graphs in terms of cycles: A graph is bipartite
iff it has no odd cycles.
• Testing whether a graph is bipartite:
Union of Graphs
• Definition:
• K4 : a union of two 4-cycles.
• Kn can be expressed as a union of k bi-partite
graphs iff n < 2k.
Eulerian Circuits
• A graph is Eulerian if it has a closed trail
containing all the edges.
• A graph is Eulerian iff it has at most one nontrivial component and all its vertices have
even degree.

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